A Key Distribution System Based on Hadamard Designs

Abstract—

The topical problem of the key distribution in a community for providing secure communication among its participants is studied. To solve this problem, key predistribution systems can be considered, where each user receives some key information they can later use to calculate independently from the system’s other participants required shared secret keys for conferences they participate in. These key distribution systems can be based on different structures, such as error-correcting codes and combinatorial designs. The weakness of such systems is the risk of collusive attacks, when abusers within the system can form a coalition and use their key information to try and calculate shared secret keys of other users. But the system guarantees the secrecy of keys provided that the number of abusers in the coalition does not exceed a threshold defined by the system structure. In this paper, we consider a key distribution system that is based on combinatorial designs, in particular, on Hadamard 3‑designs, and guarantees the secrecy of communication in case of coalitions of less than three users. New notions of combinatorial span and combinatorial rank of a subset of Hadamard code are introduced that are required for studying the system’s resilience to collusive attacks. In addition, some combinatorial properties of Hadamard codes are analyzed. The probability of a successful collusive attack against an arbitrary conference is calculated for the constructed key distribution system, depending on the capacity of the coalition of abusers.